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Elliptic curves with small embedding degree and large prime-order subgroup are key ingredients for implementing pairing-based cryptographic systems. Such "Pairing-friendly" curves are rare and thus require specific constructions. In this paper the authors give a single coherent framework that encompasses all of the constructions of pairing-friendly elliptic curves currently existing in the literature. They also include new constructions of pairing-friendly curves that improve on the previously known constructions for certain embedding degrees. Finally, for all embedding degrees up to 50, they provide recommendations as to which pairing-friendly curves to choose to best satisfy a variety of performance and security requirements.
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